More revision tutes

Is it possible to have a couple more tutes to go through the practice questions we have? I.e. maybe the usual tute time on wednesday this week and next week, and maybe a third if there is a consensus?

Optional tutorial thursday 1pm

Interaction room

Appliation of quantum hall effect in electronic devices

http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4373747

Half integer quantum hall effect in graphene

Half integer quantum hall effect in is one of the most outstanding discovery in condensed matter in the second half of the twenty first century. The difference emerges from the unique electronic properties of graphene, which reveals
electron–hole degeneracy and vanishing carrier mass near the point of charge neutrality. This unique behaviour of electrons opens up for new application in carbon based electric and magnetic field-effect technology, for example, aplication in ballistic or metallic/semiconducting graphene ribbon devices and electric field effective spin transport devices using a spin-polarized edge state.
http://www.nature.com/nature/journal/v438/n7065/abs/nature04235.html

Some sites on thermoelectric devices

http://www.novelconceptsinc.com/calculators.htm

http://www.tec-microsystems.com/EN/Intro_Thermoelectric_Coolers.html

BECs and BCS

http://jfi.uchicago.edu/~qchen/Papers/PhysRep412p1-88.pdf

This is a (lengthy) paper which looks at some similarities/differences between BECs and BCS theory for superconductors. Its a bit long to go through all of it, but the introduction is quite interesting in raising some ideas, about how BECs and BCS superconductors show properties which are two sides of the same coin.

Presentation marks

3 digits are last 3 numbers of student ID
marks are out of ten

404       5
335       5
050      5.5
658      4
459      6
344      5
172     8.5
370     7.5
677      6

Types of superconductors

superconductors is divided to two types depending on it's temperature:

1- Low temperature superconductors(LTC): it is called also conventional superconductors such as Mercury and this type characterized by low critical temperature(Tc).

2- High temperature superconductors(HTS):this type characterized by high critical temperature(Tc).for example,(La2 Ba1 Cu O4)

kondo effect and kondo lattice

Kondo effect is one of the most significant feature of heave fermion material. It is basiclly , as I said in my presentation, the increasing of resistivity which caused by the interaction between the spin of conduction electron and the spin of impurity.At zero temperature, the spin of conduction electron will make asinglit with the spin of impurity .So, the resistivity p is propotional to log (Tf/T) and in real system the resistivity will be finite.
Kondo lattice appears when we have more impurities in the system and hence the intraction between impurites is taken into account as well as the intraction between conduction electron with impurity to calculate the resistivity.Actually, now kondo effect depend on the spin of impurities (ferromagnet or antiferromagnet)and the strength btween them.

Course evaluation and optional tute

I forgot to get you to fill out the course evaluation. Sorry.
Could we do that at 1pm today in the interaction room as part of the optional tutorial?
Those who come I will make it worthwhile....

Guide for assessable presentations:

Just a useful guide for assessable presentations.
http://sydney.edu.au/health_sciences/pdfs_docs/assign_guide.pdf

Good luck all with your presentations.

Superconductivity

A pdf that includes all main points about superconductivity and their magnetic property :

http://www.imprs-am.mpg.de/summerschool2003/muramatsu_notes.pdf

Student presentations

I expect you all to attend all the presentations and to fill out the feedback and assessment forms I will provide.

I will bring the data projector but it is your responsibility to bring a laptop and to check beforehand that you can get it to work with the data projector. I will set up the data projector at least 15 minutes before class time so you can do that.

Time limits will be strictly enforced.

Pseudogap phase

At the pseudogap phase there are still conductors with different properties of the usual conductors such as, linear resistivity. Pseudogap state has some similar properties with the superconducting state, but at a very high doping and temperature they act like normal Fermi-liquid.

semiconductor and Quantum dots

It is predicted that one day it may be possible to use the array of quantum dots driven by ultra fast laser pulse to carry out quantum information proceesing. Some semiconductors with conduction band electrons with effective mass less than 1/10th of an free electron mass this maske the size of semiconductor 10nm in the quantum limit (which makes easier to adress and manipulate individual quantum dots). This size is supposed to very useful to invernt some quantum-computers with much smaller size and weight. A paradigmshift toward direct use of the quantumproperties of information processingdevices is now underway.
Useful information about the semicondutor as quantum dots can be found in;
http://www.ph.utexas.edu/classes/li/QuantumDotIntro.pdf

Archive of Semiconducting materials

Here is an archive of semiconducting elements and compounds as well as heterostructures that are based on them. It gives information such as energy gap, carrier concentration, binding energies of dopants, effective masses of charge carriers and so on.
Not much in the way of theory but I thought it looked interesting.

http://www.ioffe.rssi.ru/SVA/NSM/Semicond/index.html

Semiconductors lecture for reference

htp://videolectures.net/mit3091f04_sadoway_lec13/
A video about Intrinsic and Extrinsic Semiconductors, Doping, Compound Semiconductors, Molten Semiconductors. It's from MIT, a famous uni in the US.

Direct gap and indirect gap semiconductors

On the basis of band structure, there are two types of semiconductor: direct gap and indirect gap. The difference is that in direct gap semiconductors, the top and bottom of the valance and conduction bands are directly above each other (hence the name). In an indirect gap semiconductor however, the bottom of the conduction band is not directly above the top of the valence band.
This leads to a difference in the process to bring an electron from the valence band to the conduction band. A direct gap semiconductor only requires the transfer of energy (from say a photon) to reach the conduction band while an indirect gap semiconductor requires a change in both energy (from a photon) and momentum (from a phonon) to reach the conduction band.

problem

I tried several times today, but i still have a problem to post comments.

Doping

Generally, doping is the process of adding impurities to pure semiconductors in order to change its electrical properties. Doping in organic conductor can be chemical or electrical. There is also magnetic doping that can effect some properties like specific heat by adding small amount of impurities.

post a comment

i have the same Lan problem does anyone know how we can post a comment??

wide band gap in semiconductor

From the lecture note the energy gap in semiconductor is less than 2eV.However, i found in the wikipedia there is semiconductor that has wide band gap(more than 2eV) i couldn't understand what are the differences between this and insulator??

Impurity in semiconductor

Impurity know as donors or acceptors. The differences between them is that the former impurities provide extra electrons to the conduction band. Whereas, the acceptor impurities provide extra holes to the valence band. Also, the chemical valence helps to determine the impurity type if it is donor or acceptor. The donor impurities have high chemical valence, while the acceptors have low chemical valence.(Ashcroft and Mermin,1976)

Something's wrong

I cannot comment on any posts. Does anyone have the same problem?

An interesting guide to semi-conductors

http://britneyspears.ac/lasers.htm

For those who were a bit confused about the discussion of semi-conductors in lectures, the above link is a useful guide to their behaviour. Don't be alarmed by the URL, it is in fact a quite serious guide to semi-conductor physics, purportedly *written* by *Britney Spears*.

It has sections on the basics of semi-conductors, junctions, recombination and the other topics we touched on. I found this part on p-n junctions particularly useful.

http://britneyspears.ac/physics/pn/pnjunct.htm

The p-n junction

one of the most important application in semiconductors is tje p-n junction. when p(holes)-type and n(electrons)-type are joined the current will move in only one direction and not revers.
to see a clear picture about the p-n junction see this link
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/pnjun.html

Semiconductor and Band-gap

Some of the semiconductors like silicon (Si) and germanium (Ge) have band gap energy of 1.1 and 0.7 eV respectively, they absorb visible radiation. from the statical law of thermal energy distribution in solid, at room temperature there are significant numbers of electrons in valance band which acquire sufficient energy to cross the energy gap, this is semiconductivity. Unlike in metal, semiconductivity increases with the temperature not with the increased lattice vibration. The properties of semiconductor depends on the number and types of charge carriers n-type (negative- electrons), p-type (holes, positive) and can be controlled by the apropriate dopant elements. For more information;
http://ocw.mit.edu/courses/materials-science-and-engineering/3-091sc-introduction-to-solid-state-chemistry-fall-2010/syllabus/MIT3_091SCF09_aln03.pdf

question

i am confused about the physical meaning of j(j+1),s(s+1),and l(l+1) in paramagnetism and diamagnetism in chapter 31 in Ashcroft and Mermin
please can any one explain it ?

Course summary

You can hand this to me on friday June 24 but I will not start marking them until monday June 27.

Read the course profile for more information for what the summary should involve.

The summary should sent to me via email as a pdf file.

Since there have already been several incidents of plagiarism in this course, all the summaries will be run through the program Turnitin.

Tutorial today at 2pm

this will be open office hours
I will be available in the interaction room to answer questions about lectures, past tutorial and exam questions, your papers for presentations, ...
you might also try answering the problems for chapter 33 and 34 of ashcroft and mermin.

Updated lecture notes

You can download the latest version of my notes here.

Semiconductor Notes

Today's lecture notes are available for download here.

Paper presentation schedule

Here is the proposed schedule. You are welcome to swap with one another.
Each presentation will be 12 minutes plus 3 minutes for questions.
Time limits will be rigidly enforced.

The material below is from John Wilkins one page guides and should be read and applied before giving your talk.

• Preparing a talk. [html ], [postscript ]
• Talk Grit [html], [postscript] (Jim Garland)
• How to give a Terrible Talk 
• Viewgraphs Preparation: NOT 

----------
Monday May 30 2pm

Robert

FLUCTUATING VALENCE IN A CORRELATED SOLID AND THE ANOMALOUS PROPERTIES OF -PLUTONIUM

Shishir

THE BIRTH OF TOPOLOGICAL INSULATORS

Negar

Quasiparticles at the Verge of Localization near the Mott Metal-Insulator Transition in a Two-Dimensional Material

---------

Tuesday May 31 11am

Sam

A UNIFIED EXPLANATION OF THE KADOWAKI–WOODS RATIO IN STRONGLY CORRELATED METALS

Thurayana

TUNABLE FRÖHLICH POLARONS IN ORGANIC SINGLE-CRYSTAL TRANSISTORS

Lan

EXPERIMENTAL OBSERVATION OF THE QUANTUM HALL EFFECT AND BERRY'S PHASE IN GRAPHENE
-------
Wednesday June 1 noon

Josh

COMPLEX THERMOELECTRIC MATERIALS

Saeed

QUANTUM CRITICALITY IN HEAVY-FERMION METALS

Shahd

FLUCTUATING SUPERCONDUCTIVITY IN ORGANIC MOLECULAR METALS CLOSE TO THE MOTT TRANSITION

Updated lecture slides on superconductivity

are available here.
Again, reading chapter 34 of Ashcroft and Mermin is highly recommended (essential).

Flying frogs and levitating magnets with your hand

With all the assignments/speeches we have to sort through at the moment I though this would be a pleasant diversion on a Monday morning:

http://www.physics.ucla.edu/marty/diamag/diajap00.pdf

The paper is based on Earnshaw's theorem, which proves that there exists no stable equilibriums for 1/r^2 forces. In the simplest context, it explains why no matter how hard you try you've never been able to balance two magnets with their opposing poles as a kid.

The paper discusses how pre-WW2 it was shown by Braunbeck that it was possible to achieve stable equilibria with diamagnetic materials (hence why superconductors can levitate). They apply some of his basic theory to some very interesting examples, i.e. finding the stable equilibrium point to levitate a frog in mid-air. This is accomplished due to the dia-magnetic nature of many molecules, such as water and proteins. They even show how the minute diamagnetic nature of your fingers (and apparently a book on the Feynmann lectures) can be used to stabilise the equilibrium point of a small magnet in a field.

High temperature superconductivity

A critical temperature of 30K is the theoretical limit of the BCS theory of superconductors. However, some materials have been found to have Tc as high as ~130K (with one claimed as high as 164K). These materials cannot be explained by BCS theory although paired electrons may still be involved in at least some of these cases.

High temperature superconductors is currently one major problem within theoretical physics that remains unsolved (for now) and theory surrounding them is an active area of research.

http://adsabs.harvard.edu/abs/1993Natur.365..323C
http://prb.aps.org/abstract/PRB/v50/i6/p4260_1
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/hitc.html#c1

Confusing about supeconductivity devices

Superconductor exists in an acociatate with a critical temperature Tc. In fact, temperature does not stay at the same scale forever unless it is in lab condition. It means for every devices having supeconductivity character, they have to include a temperature adjuster to keep them always in Tc. I wonder whether my assumption is true?

Meissner Effect and Superconductivity

Meissner Effect:

When any superconductor is cooled below its critical temperature Tc, the external magnectic field is expelled.

some more useful informations can be found in this link.

http://www.phi.kit.edu/ee2/abb//Superconductivity-03.pdf

Superconductor

"The best information online about
superconductivity."
- Energy Science News
http://www.superconductors.org/

Magnetic structure

There are three different kinds of magnetic which are:

1. Ferromagnetic: In ferromagnetic all the moments are in the same direction of the spontaneous magnetisation.

2. Antiferromagnetic: In antiferromagnetic all the moments are in opposite direction of the spontaneous magnetisation and give the total zero moment.

3. Ferrimagnetic: In ferrimagnetic there are a mixture of moment in opposite direction but give the total non-zero moment.

Limit of superconductivity

Destroying superconductivity in a material requires the breaking of the cooper pairs. There are at least two ways to do this: heat the system to the point that the thermal energy is equal to the binding energy of the cooper pairs,
or the absorbtion of electromagnetic radiation of frequency w such that hbarw ~binding energy.

Superconductivitey

Superconducting phenomenon was observed in 1911 by the German scientist Onnes through his experiments on electrical conductivity of metals at low temperatures.He observed that the resistivity of the pure mercury disappeared suddenly at low temperature (4.2 K) this temperature ,which has this property,called critical temperature and above this temperature the resistivity is small and metal is in normal condition .

Energy gap in superconductor

For a normal metals there is a Drude peak, but the important point about the energy gap in superconductor is, some say that the condition for superconductivity is having the energy gap which is wrong. the energy gap is not a necessary condition for superconductivity.

Magnetism due to unfilled atomic shells

Magnetism in isolated atoms and ions can be due to the electrons of unfilled atomic shells. Although nuclei have a magnetic moment, it is usually ignored because it is much smaller than the magnetic moments of the electrons in the unfilled shells. In an isolated atom, the total magnetic moment for the atom is the sum of the moments of each electron. That can be found by summing the magnetic moment of the orbital ML and the moment of the electron spin Ms. These two moments will vanish for filled shells, so only the unfilled shells moment is count in the total magnetic moment.

Draft slides on superconductivity

Here is the current draft version of my Slides for the lectures on superconductivity.
But, they are no substitute for reading chapter 34 of Ashcroft and Mermin.

Antiferromagnetism

An antiferromagnetic interaction acts to anti-align neighboring spins. Antiferromagnetic interaction exists where energy J < 0, where J is the sum over all pairs, i, j, of an interaction term J(i, j), times the spin of atom i times the spin of atom j. J > 0 indicates a ferromagnetic interaction. The combination of both can lead to spin glass behavior.

When no external field is applied, the antiferromagnetic structure corresponds to a vanishing total magnetization. In a field, a kind of ferrimagnetic behavior may be displayed in the antiferromagnetic phase, with the absolute value of one of the sublattice magnetizations differing from that of the other sublattice, resulting in a nonzero net magnetization.

The magnetic susceptibility of an antiferromagnetic material typically shows a maximum at the Néel temperature. In contrast, at the transition between the ferromagnetic to the paramagnetic phases the susceptibility will diverge. In the antiferromagnetic case, a divergence is observed in the staggered susceptibility.

Various microscopic (exchange) interactions between the magnetic moments or spins may lead to antiferromagnetic structures. In the simplest case, one may consider an Ising model on an bipartite lattice, e.g. the simple cubic lattice, with couplings between spins at nearest neighbor sites. Depending on the sign of that interaction, ferromagnetic or antiferromagnetic order will result. Geometrical frustration or competing ferro- and antiferromagnetic interactions may lead to different and, perhaps, more complicated magnetic structures.

Superparamagnetism

Doing some reading I stumbled onto an interesting magnetic behaviour. Beyond our discussions of ferromagnetism, paramagnetism etc there exist more complex states of magnetic order/disorder. In particular there is a state known as superparamagnetism. This occurs in small ferro- or ferrimagnets, and is defined by flips in the overall spin state of the system due to temperature. The ordering of the system is not lost, only the orientation of the ordering (i.e. all spins may flip from up to down but retain <|S|> = 1).

This flipping is characterized by the Neel relaxation time:

tau_N = tau_0*exp ((K*V)/(k_B*T))

Where K is the particles magnetic anisotropic energy. The relaxation time may range from nanoseconds to years for particles. However over a time scale much greater than tau_N the net magnetization will be measured to be zero.

The above was assumed to be in zero magnetic field. However if we now apply a field to the superparamagnetic particle it will order accordingly, much like a paramagnet. Due to this ordering, the particle can be said to have a magnetic susceptibility. This magnetic susceptibility will be far larger than a normal paramagnet hence the term superparamagnet.

This state of magnetism is of great importance to technology, as it is a limiting factor for storage efficiency on magnet-based hard drives. Superparamagnetism sets a lower limit on the particle size which may be used to store information, as the relaxation time is proportional to exp(V).

Assignment 5 - due Monday May 30

You can get the latest assignment here.

In tomorrow's tutorial I suggest we do more practice questions. I have uploaded past exam papers from 2005 and 2007 that we can work on, but feel free to bring any other questions that you need help with.

Updated lecture notes

The latest version of my notes are available to download here.

Course schedule until end

Week 11
Tuesday 11am - Paul - magnetism
Wed. noon - Paul - tute
Wed. 2pm - Ross - lecture - superconductivity - read ch. 34 in Ashcroft and Mermin
Thursday 11am - Ross - optional problem solving session

Week 12

Monday 2pm - Ross - lecture - superconductivity - read ch. 34 

Tuesday 11am - Paul - lecture - semiconductors - read ch. 28

Wed. noon - Paul - lecture - semiconductors - read ch. 29

Wed. 2pm - tutorial

Thursday 1 pm - Ross - optional problem solving session

Week 13

student presentations

Energies of Singlet and Triplet States

Triplet spatial wavefunction is zero when the two electrons are at the same position, wheras,
the singlet wavefunction is nonzero. Because the electrons repel each other more when they are close to one another, we therefore expect the singlet to have more electronelectron repulsion and a higher energy. This rule turns out to hold quite generally and is called Hund’s rule : for degenerate noninteracting states, the configuration with highest spin multiplicity lies lowest in energy. Hence, triplet are expected with lower energies. more can be found on this site which may be helpful to understand soem more topics discussed in class.
http://ocw.mit.edu/courses/chemistry/5-61-physical-chemistry-fall-2007/lecture-notes/lecture26.pdf

Quantum Heisenberg Model

http://en.wikipedia.org/wiki/Heisenberg_model_%28quantum%29

The above is a link to a full description of the Quantum Heisenberg Model as we derived in class. Note that in class we considered the case of zero external field and unlimited neighbor interactions. Usually the Heisenberg model is referred to when considering a lattice with nearest-neighbor interactions only. The model is generally solvable for the ground-state, however as most of us are finding in Phys4040, it is not simple.

In general the Heisenberg model is much simpler when solved computationally, and in fact is much more efficient computationally than the classical Ising model. Using a suitable eigenvalue technique, for instance the Lanczos algorithm, it is possible to calculate the ground-state and lower excited states for an NxN lattice (and higher dimensions, although memory can become a problem for any lattice size which is a reasonable approximation to the thermodynamic limit). An important consequence one finds upon solving, is that the ferromagnet and anti-ferromagnet arrangements seen in class only occur for zero field. When a magnetic field is apply, a superposition of different spin states is found.

Optional problem solving session on thursdays at noon

For the next five weeks at thursday I will be in the interaction room from noon-1pm to help you with any problems you want to work through.

Notes on magnetic ordering

You can find the notes for today's lecture here. I would also recommend reading Chapter 32 from Ashcroft & Mermin beforehand.

Ions with a partially filled shell

Ashcroft Mermin p.650. Assuming that we have a free ion with a partially filled shell. Why did not electrons interact with one another, the ionic ground state will be degenerate? Please help!

Bohr magneton calculation

It is known that Bohr magneton is defined as (e.h_bar)/(2.m.c) where e is electron charge, h_bar is Planck's constant, m is electron rest mass. I'm just not sure about c. Is it the speed of light in vacuum? When calculating it, c is assumed to be 1, the same as in general relativity. Can anyone help?

Question about assignment 4

Can anyone find out how to get the factor of 2 in q2 for m*/m = UG/2E0(G/2)? becuase like the tutorial again I got m*/m = 2UG/E0(G). In the tutorial we also got E0(G) instead of E0(G/2), and I got the same result as tutorial again. I think maybe we should follow some other way. can anyone help me with it?

Kadowaki–Woods ratio

The Kadowaki–Woods ratio is the ratio of A, the quadratic term of the resistivity and γ2, the linear term of the specific heat. This ratio is found to be a constant for transition metals, and for heavy-fermion compounds, although at different values.

In 2009, Anthony Jacko, John Fjaerestad, and Ben Powell showed that the different ratios could be understood on the basis of different materials specific properties, such as the density of states and the electron density, even before electron-electron interactions were taken into account.

Fermi liquid theory

Landau's Fermi liquid theory can explain that why the independent electron approximation works, but it fails when there is a strong interaction between electrons, in one and two dimensions (where there are strong interaction between electron for example in cuprate materials), for high Tc and for heavy fermions near a quantum critical point.

Hund's Rule

Developed by the German scientist, Friedrich Hund (1896-1997), Hund's rule allows scientists to predict the order in which electrons fill an atom's suborbital shells. Hund's rule is based on the Aufbau principle that electrons are added to the lowest available energylevel (shell) of an atom.

Around each atomicnucleus, electrons occupy energy levels termed shells. Each shell is identified with quantum number, n, that defines the mainenergy level. Each main level is made up of a number of sublevels. These sublevels are identified by their shapes: s sublevels have 1 orbital, p sublevels have 3 orbitals, d sublevels have 5 orbitals; and f. sublevels have 7 orbitals. Each orbital can contain only 3 electrons spinning in opposite directions .

Although each suborbital can hold two electrons, the electrons all carry negative charges and, because like charges repel, electrons repel each other. In accord with Hund's rule, electrons space themselves as far apart as possible by occupying all availablevacant suborbitals before pairing up with another electron. The unpaired electrons all have the same spin quantum number (represented in electron configuration diagrams with arrows all pointing either upward or downward).

The Pauli exclusion principle states that each electron must have its own unique set of quantum numbers that specify its energy. Accordingly, because all electrons have a spin of 1/2, each suborbital can hold up to two electrons only if their spins are paired +1/2 with -1/2. In electron configuration diagrams, paired electrons with opposite spins are represented by paired arrows pointing up and down.

For example, if there are three available p orbitals (p^x, p^y, p^z) the first three electons will fill these one at a time, each with the same spin. When the fourth electron is added, it will enter the (p^x orbital and will adopt the opposite spin since this is a lower energy configuration.

Although Hund's rule accurately predicts the electron configuration of most elements, exceptions exist, especially when atoms and ions have the opportunity to gain additional stability by having filled s orbitals or half- filled or filled d or f orbitals.

Diamagnetism and Paramagnetism notes

Here are some notes on diamagnetism and paramagnetism from the national university of taiwan.
They cover similar areas to our lectures but also mention and includes crystal field splitting and nuclear demagnetisnation.

http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/chap11.pdf

Nuclear Diamagnetism

While reading about Diamagnetism, I came across something called nuclear diamagnetism. Which is similar to diamagnetism except that the external field effects the nucleons of an atom. I couldn't seem to find much else on it but I expect that since atomic nuclei are usually screened by electrons, this is only relevant for ions that have been stripped of most or all of their electrons such as plasmas.

Diamagnetism

The most important keys for diamagnetism are that the materials are very weakly respond to magnetic fields. Also,the atoms or molecules of diamagnetic materials contain no unpaired spins and the electrons orbiting in atoms oppose the action of the external field as with Lenz’s law. lastly.diamagnetism is not affected by changes in temperature and the value of the susceptibility is always negative.

Magnetisation

we understood from the lecture that magnetisation is " a measure of how a material act when a magnetic field is applied to it" . Also, i found this website that includes many information about magnetisation and answered in more detail the question of what happen to an atom/ion in a uniform magnetic field?

http://physics.unl.edu/~tsymbal/teaching/SSP-927/Section%2015_Magnetic_Properties_1.pdf

Updated lecture notes

The latest version of my notes is available here.

Instabilities of Fermi liquid

I found some interesting informations related to instabilities of fermi liquid.
Generally, Fermi liquid theory is very successful for single particle excitation in solids, but also it can have some phase transition into another state with some degree of order. Susceptibility is responsible for the phase transition in Fermi liquid. Some perturbation h is present in Fermi liquid which give the variation in O (charge, spin, current etc.), then (q,E) = xi(q,E) h(q,E), q = charge, E = energy. For E = 0 we observe the static perturbation and static response, so if there is increase in susceptibility then the small amount perturbation h can also effect the process. At a critical temperature Tc = 0 the susceptibility diverges so in this case a small perturbation would induce a finite response in the system. Now if the calculation is made over the susceptibility, there is prediction over the Fermi liquid phase, with symmetry, where the system wants to go with the change is temperature. This means the various quantities can diverge with the lower temperature (Tc). Lower the critical temperature the ground state breaks the symmetry. Hence the calculation made over the susceptibility does not hold any accuracy.

Some papers for student presentations

Here is a preliminary list of possible papers you can use for your presentation.
First come, first served.
Claim yours with a comment below.
I welcome alternative suggestions.

You will have to give a 15 minute presentation where you
-summarise the key ideas and results of the paper
-relate the contents to what you have learnt in the course
-state things you did not understand
-any weaknesses you see in the paper

Marks will be based on
-quality of presentation
-level of understanding of the paper
-ability to relate the paper to what you have learned in the course
-ability to answer questions

Experimental observation of the quantum Hall effect and Berry's phase in graphene

Ideal diode equation for organic heterojunctions. I. Derivation and application

Complex thermoelectric materials

Tunable Fröhlich polarons in organic single-crystal transistors

Understanding ion motion in disordered solids from impedance spectroscopy scaling

Quasiparticles at the Verge of Localization near the Mott Metal-Insulator Transition in a Two-Dimensional Material

The birth of topological insulators

Fluctuating valence in a correlated solid and the anomalous properties of -plutonium

Fluctuating superconductivity in organic molecular metals close to the Mott transition

Quantum criticality in heavy-fermion metals

A unified explanation of the Kadowaki–Woods ratio in strongly correlated metals

Magnetism lecture notes

My current notes for tomorrow's lecture can be downloaded here.

Fermi Liquid Theory and the Independent electron Approximation

In the lecture, the argument was made that at T>0K, the scattering rate:

1/tau = A(E1 - Ef)^2 + B(Kb.T)^2, where A, B are T-independent constants.

So the scattering time tau is approximately proportional to T^2. Arguments are made in A&M that the scattering time tau depends on the interaction potential from Thomas Fermi screening: 4.pi.e^2/k_0^2.

Performing quite a rough dimensional analysis on this quantity, they get that:

1/tau ~ (Kb.T)^2 / h-bar.Ef

At room T, tau is of order 10^-10 seconds, which is 4 orders of magnitude longer than the typical scattering due to impurities. This suggests that the e-e interactions do not have a significant effect on the validity of the independent electron approximation.

Slides for lecture on Fermi liquid theory - hard copies in lecture

Getting the most out of lectures

The lectures closely follow Ashcroft and Mermin. I suggest that before each lecture you read the suggested part of the text.
The lecture is designed to highlight the key ideas, concepts, equations, and experimental results. There is insufficient time to explain everything in detail, particularly to go through every step of the algebra in every derivation. If you are so inclined you should do this by yourself.
There is also no point in me writing out lectures notes which just say the same thing as what the text says.
You should then re-read the relevant part of the text after the lecture.

Todays lecture will be on Landau's Fermi liquid theory of metals and quasi-particles. The relevant parts of the text are pages 345-351.

My final lectures will be on Superconductivity, covering all of chapter 34.

Question about Mid semester exam

Can anyone tell me why there is not any factor of 1/a^2 in question 2 part b for g(E)?

Quasiparticle

Quasi particles can be defined as a compensation of a particle and its effect on the surrounding. A good example is holes: which are the missing of electrons, and that cause holes to be carriers for positive charges. Quasi particles approach is important in simplifying the many particles Schrodinger equation.

Hartree Approximation

Schrodinger equation can become very complex if the electron-electron interaction was included. So, to simplify the equation and actually solve it, Hartree Approximation is used to introduce the molecular orbital approximation and make the particles more independent.

Jellium model

I found a note that's somehow interesting to me. In the note, the Screening of the Coulomb interaction, Friedel oscillations, plasmons (collective oscillations)
are explained very detailed.
people.web.psi.ch/mudry/FALL01/lecture03.pdf

The basic eq of nonlinear Thomas-Fermi theory

While reviewing chap 17 (Ashcroft/Mermin), I cannot get the basic eq of nonlinear Thomas-Fermi(17.46) by combining (17.44) and (17.45). Can anyone help?

Some links regarding Hartree-Fock

Here some links about the Hartree-Fock method.

http://www.physics.uc.edu/~pkent/thesis/pkthnode13.html

http://www.chm.davidson.edu/ronutt/che401/HartreeFock/HartreeFock.htm

SI Units

Ross has been reminding us recently about the need for us to use units in calculations. He has a good point, but I for one have not sought to remember the necessary conversions between various different units, like J = kg.m/s^2, as I've always calculated in straight SI units and put the units on at the end. In a discussion about it, another student pointed out this is in fact an incomplete approach, as it turns out there are only 7 fundamental units. They are:

metre for length
kilogram for mass
second for time
ampere for electric current
kelvin for temperature
candela for luminous intensity
mole for the amount of substance.

I thought this was quite interesting, not just the fact that a joule is in fact technically not an SI unit, but more for the choice of ampere for electric current. This means technically a charge can be measured properly as A.s and not C. I would argue that C is much more fundamental, as if you change reference frames a current may disappear, but a collection of charge will not. Anyone know why ampere's are chosen, or disagree with my choice of C?

Plasmons

Plasmons are about to be brought up in the lecture material, but I stumbled across them in some reading of the Jellium model. They are a quasi-particle brought about by oscillations of the free-electron density against the positive ions in a metal.

An interesting consequence of plasmons is the optical properties of a metal. The frequency of the density oscillation is known as the plasma frequency. When light above the plasma frequency is incident on a metal it is transmitted, whereas light below this frequency is reflected due to the collective screening behavior of the electrons.

Paper presentation

I'm wondering about the paper presentation.

In regards to the presentation, what guidelines are there besides a (close) connection to the course? Are recent papers expected as opposed to older ones?

Question

Does anyone answer correctly part four of question two in the four assignment that we have done?if yes please, present it here?

Pauli Principle and e-e interaction

1. Fermi sphere cannot hold two electrons due to energy conservation and Pauli principle. So one of them must be outside the shpere for scattering.
2. Electrons can scatter itself at finite temperature, dE1 = KT = 1/40 eV at room temperature. For Ef = 2.5 eV, 10exp-4 electrons have chance for scattering.
3. Scattering rate tou ^-1 is directly proportional to T^2, low temperature and pure sample eliminates the thermal and impurity scattering so relatively good e-e scattering can be observed.
4. Fermi surface is stable because of Pauli principle.
Fell free to add some more points related e-e scattering.

jellium model definition

I found the definition of the jellium model that is "A model of electron-electron interactions in a metal in which the positive charge associated with the ion cores immersed in the sea of conduction electrons is replaced by a uniform positive background charge terminating along a plane that represents the surface of the metal"
http://www.answers.com/topic/jellium-model.

Limitations of Hartree Approximation

1. The Pauli principle doesnot hold good for many body approximations.
2. The total energy calculation comes to be positive i.e. electron gas is unstable.
Are there some more limitations?

Thomas-Fermi Screening

A basic illustration of screening involves considering a positive ion in an electron gas, which will aggregate electrons around itself. The presence of this negative charge surrounding the ion reduces the electric field of the ion; this is called screening.

The Thomas-Fermi theory of screening considers a slowly varying (in r) potential. The modification in the electron number density due to the presence of this potential is then calculated, as in the lectures. From this, after some maths, the total potential as a function of r, which is found to decay exponentially in r. The distance at which the charge is effectively screened (becomes negligible) is of the same order as the inter-atomic spacing, meaning the effect of the charge is not long ranged at all.

Energy levels of Helium atom

I found this websit when i search for explaination of Helium energy level. It might help.


http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html

Optional extra tutorial

Some students have asked for extra feedback and advice on passing the course.
Tomorrow at 1pm I will be available in the interaction room. If you wish to work through tutorial problems and/or past exam questions, and get feedback I will be available then. If it is popular enough I will do this every week, at a mutually agreeable time.

Exam practice and feedback

Several students have asked for concrete suggestions on improving their performance on the final exam. I suggest the following exercise.

Write out a new formula sheet.
Take the mid-semester exam and try and do it again in one hour by yourself.
Give me your written solutions and I will mark it as I would normally.
Note this will not change your mark, but hopefully it will help you get a better grasp of the material, and get more feedback of what I expect.

Hartree Approximation

The Hartree approximation arises from a first attempt to account for the effects of electron-electron interactions, which are neglected when using the independent electron approximation (as in the bloch model etc). The single-electron wavefunction seems insufficient to properly describe how the electrons in an N-particle system interact. So the Schrodinger equation for an N-particle system is considered, i.e. we want a wavefunction for the N electron system. However, attempts to solve this equation prove to be futile.

Instead, the single electron wavefunction that best represents the interactions is analysed, specifically, a potential that includes the potential of the ions U-ion and the electric fields from the other electrons in the system U-el (the approximation is made that this is given by a smooth distribution of negative charge). The contributions from all the electrons to the potential energy of the electron considered can then be worked out (A&M pg. 330-331) and plugged into the Schrodinger equation for that particular electron. If such an equation is written for each electron, the set of equations is known as the Hartree Equations. The Hartree equations are solved computationally by proposing a form of the potential U-el, solving the equation to obtain the wavefunction, from which the next U-el is found, and so forth until the potential reaches a certain degree of accuracy (does not change very much between successive iterations).

Energy bands

Here is a useful Pdf for better understanding of energy bands and periodic potential.

http://www.teknik.uu.se/ftf/education/ftf1/forelasningar/overview/energybands.pdf

Lecture slides on electron-electron interactions

Here is are the draft slides for the next few lectures.
At the tutorial on wednesday I will work through the answers to the mid-semester exam.

NFE model

Here is a very useful pdf for a nearly free electron model. (NFE)

http://www.cmmp.ucl.ac.uk/~ahh/teaching/3C25/Lecture19s.pdf

Summary of tight binding model

As it appears from the model name, in this approximation electrons are tightly bind to their atoms with limited interaction with other atoms in their neighbor. this approximation minimise the overlap. but small correction in Hamiltonian is needed.

Semimetal and Semiconductor

Ashcroft and Mermin
At T = 0, a pure semimetal behaves as good conductor, as there are some partialy filled electrons and holes band.
At T = 0, a pure semiconductor behaves as insulator, as the carriers are either due to thermal excitation or presence of impurities.

Fermi Surface

The fermi surface of Lithium is not known. What is martensitic transformation ? Is it like not having a proper crystal structure at low temperature? Can anyone help me ?

Reading for next week

Beginning next tuesday I will lecture on "Beyond the independent electron approximation", closely following chapter 17 and pages 674-9 of Ashcroft and Mermin.
Key concepts include:
The many-body Schrodinger equation
Hartree approximation and the self-consistent field
Hartree-Fock approximation and the exchange interaction
Screening
Fermi liquid theory
Singlet-Triplet splitting

How are predictions derived from fermisurface ?

As we know that the is an boundary separated the states that are occupied and unoccupied. Moreover, its shape is derived from the periodicity and symmetry of the crystalline lattice. It's said that, for these characters, Fermi surface is useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. Although I have read carefully Bloch model, band energy, electron on periodic and weak potential, I cannot understand how such prediction can be derive from Fermi surface? Any one can help?

Fermi surfaces

Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields H, for example the de Haas-van Alphen effect (dHvA) and the Shubnikov–de Haas effect (SdH). The former is an oscillation in magnetic susceptibility and the latter in resistivity. The oscillations are periodic versus 1 / H and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by Lev Landau. The new states are called Landau levels and are separated by an energy where ωc = eH / m * c is called the cyclotron frequency, e is the electronic charge, m * is the electron effective mass and c is the speed of light.

Landau Levels

I found this useful for better understanding of the Landau levels and how to get quantum Hall effects out of it.

http://landaucongress.itp.ac.ru/Talks/halperin.pdf

Necessary condition for observing the quantum oscillation

For observing the quantum oscillation three factors are important:

1. High magnetic fields

2. Low temperature

3. Clean samples which means having pure crystals

use of superconductor in train

Ever since the discovery of superconductors, there has been great interest in their use in electronics. It turns out that the magnetic properties of superconductors has been more useful in a larger variety of applications than the lack of resistance.

Maglev trains use superconductors to levitate the train above magnetic rails. This enables them to operate without friction, and therefore acheive unheard of speeds. The maglev train below is being installed at the Old Dominion University in Hampton, Virginia. It is the first to be installed in the US. Unfortunately, due to the short track it is on, it can only reach speeds of 40 miles per hour. Maglevs, with sufficient track, can reach speeds over 300 mph. A new Maglev train in Shanghai recently broke the 500 Km/h barrier (310 mph). These trains are also more efficient because there less energy loss to friction between the train and the track.

Visualising Landau level filling and the IQHE

I have updated the slides for todays lecture. On my blog there is a post which has a link to the simulation I showed.

de Haas van Alphen Effect

http://phy.ntnu.edu.tw/~changmc/Teach/SS/SSG_note/grad_chap14.pdf

This is a nice little collection of info on the de Haas van Alphen effect I found (might be where Ross pinched some pictures from).

For an example of how you can actually apply the effect to construct the Fermi surface, have a look through this paper:

http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-14075.pdf

If you don't want to get into the gritty details at least check out fig. 2 to look at the relations between measurement and theory.

Lecture on determination of the Fermi surface

Here are draft slides for some of todays lecture which is closely related to Chapter 14 and the beginning Chapter 15 in Ashcroft and Mermin.

Notes on semi-classical transport theory

The notes I will use in today's lecture follow chapter 12 of Ashcroft and Mermin closely.

New Organic Conductor for Electronic Devices

A new organic polymer has been developed. It can be laid down using simple printing techniques rather than the expensive and elaborate methods used to process silicon, 6 times faster than previous organic polymers.

Now just as fast as silicon plus much cheaper, this inexpensive organic conductor could be used in areas where silicon struggles to compete, eventually slashing the cost of transistors, PDA’s, flat panel screens and bringing electronic paper into common use.

The Development of Organic Conductors

I found a link that may be useful for someone need to know more about Organic Conductor. This has 5 luctures: Lecture 1. the Development of Organic Conductors
Metals, Superconductors and Semiconductors. Lecture 2A. Introduction and Synthesis of Important. Conjugated Polymers. Lecture 2B. Solid State Polymerization
Lecture 3. Fullerene Chemistry. Lecture 3B. Molecular Engineering. www.lios.at/VirtualAcademy/.../Organic_Conductors_Intro_Lecture.pdf - Similar

Filled bands are inert.

The consequences of this are:

1. having full bands in insulators.

2. only the partially filled band supply the current in a metal.

3. electron current adding to the hole current is equal to zero.

Huckel assumption

Huckle method is based on two important points:

1. we only consider the pi orbitals.

2. valence electrons move in the potential field of the inner shell electrons and nuclei.

Huckel Method

Here are some lecture note from another lecture on the Huckel method. Some examples are shown including the benzene ring.

http://ocw.kyushu-u.ac.jp/3063/0002/lecture/QC_AH3_LM2.pdf

Practice Mid-Semester Question

Thought I'd throw this out there to see if anyone can help.

Has anyone had success with question 2 iv) of the practice mid-semester? I know that:

h^2/2m* = d^2E/dk^2

and have tried a few other methods, but I can't quite seem to get the answer. Any suggestions? I'm figuring I'm probably not taking the approximation near the boundary correctly.

Delocalisation of electrons

i read one article that said "Delocalisation is when electrons are not associated with one atom but are spread over several atoms.So the electrons are not directly bonded with any atoms but effectively 'float' above and below the molecule in electron clouds".This means In the case of hydrocarbons, delocalisation occurs in Benzene rings, where a hexagon of six carbon atoms has delocalised electrons spread over the whole ring.For metals, electrons are delocalised over the whole crystal structure and outer electrons of the metal atoms are shared in an electron sea, and are not confined to particular atoms.

Molecular Orbital Theory

In our discussion of organic conductors, the notion of molecular orbitals was introduced and used in the Huckel Method. It was interesting that the electrons were treated not as dependent on a single atom, but rather being shared over the whole molecule, that is to say, the electron's motion is influenced by all nuclei in the molecule.

Each such molecule has a set of molecular orbitals, which are simply linear combinations of the atomic orbitals, and the coefficients of these sums (in the linear combination) are found using the Huckel method.

Advantages of organic electronics

One of the main advantages of organic electronics is the possibility of manufacturing components and circuits over large areas, while silicon chips are restricted to the area of circular pads of limited sizes.

Most of these advantages, however, is certainly the fact that organic electronics can be fabricated on plastic substrates, thin and flexible.

Only the flash memory transistor, the silicon component found in the pen-drives in digital cameras and MP3 players, continued to resist the benefits of plastic.

Now, Physicists built the first organic transistor Flash memory plastic.

Huckle Method.

A simpler explanation of the Huckle Method, http://www.jce.divched.org/JCEDLib/LivTexts/pChem/JCE2005p1880_2LTXT/QuantumStates/Bookfolder/L21Huckel_MO_Theory.htm

Tutorial solutions

Tute 1
Tute 2
Tute 3

Reading for next week

Ashcroft and Mermin, Chapter 13
The semi-classical theory of conduction in metals

Important equations are
13.11, 13.19
13.22-25
13.44-46
13.58, 13.61

Magnetoresistance for Open Orbits

A good feel for what saturation/non-saturation of magneto-resistance represents can be found around p235-239 in Ashcroft and Mermin.

For the open orbit case we discussed in the lecture the form of the current:

j = sigma(0)n(n.E) + sigma(1).E

Where n unit vector in the direction of the open orbit. It was argued that the component sigma(1) vanishes in the high-field limit, whereas the sigma(0) contribution does not. This means that unlike the closed-orbit case the magneto-resistance will not approach a constant.

To see this last assertion it is useful to examine a simple case where in the high-field limit the current flow is directed in a direction =/ n, say n'. From our statement above this means that the projection of the electric field, E.n = 0.

Following the working in Ashcroft and Mermin, this gives a particular form for the electric field which I will not write here. Suffice to say, defining the magneto-resistance rho as;

rho = E.j/|j|

One can arrive at the expression:

rho = (n'.j)^2/(n'.sigma(1).n')

Now as we have previously argued that sigma(1) vanishes in the high-field limit we clearly see that the magneto-resistance must diverge, i.e. it grows with increasing magnetic field.

Lecture notes

You can download a copy of my notes for today's lecture on the similarities between the tight-binding approximation and the Huckel model here.

Magnetoresistance in orbits

I tried to find out why magnetoresistance in open orbits is higher than it in closed orbits? If any one have the answer please, display it here

Semiclassical Electron Dynamics

The semiclassical model of electron dynamics sets out to describe the behaviour of Bloch electrons between collisions. In order to have energy En(k) that is approx constant over all Bloch levels in a wave-packet, the spreading of the wavevector must be small compared to the 1st Brillouin Zone: Δk << |G| ~ 1/a, where G is a reciprocal lattice vector and a is the lattice constant. This implies that Δk >> a, meaning the wave-packet is spread over many primitive cells in real space.

The semi-classical model looks at the effect on the electron dynamics of external electric or magnetic fields that are approximately uniform over the scale of one such wave-packet. The periodic potential considered in the Bloch model is responsible for the important distinction between the classical and semi-classical models (because the free electron approximation is no longer valid). This potential changes on a scale far smaller than that of the wave-packet, requiring a quantum mechanical treatment.

Hall Effect

If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers which tends to push them to one side of the conductor. This is most evident in a thin flat conductor as illustrated. A buildup of charge at the sides of the conductors will balance this magnetic influence, producing a measurable voltage between the two sides of the conductor. The presence of this measurable transverse voltage is called the Hall effect after E. H. Hall who discovered it in 1879. Note that the direction of the current I in the diagram is that of conventional current, so that the motion of electrons is in the opposite direction. That further confuses all the "right-hand rule" manipulations you have to go through to get the direction of the forces. The Hall voltage is given by VH = IB/ned

Show The Hall effect can be used to measure magnetic fields with a Hall probe.

More slides on "http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/hall.html"

Revised schedule for this week

Today 11am
Ross McKenzie lectures on transport in the Bloch model

Wednesday noon
Ross gives tutorial looking at old mid-semester exam questions

Wednesday 2pm
Paul Shaw lectures on the relationship between tight-binding and the Huckel model in chemistry.

Assignment 4: due Tuesday May 3

You can download the assignment here.

A scan of the relevant pages from Solid State Simulations can be found here.

Slides for electron transport in the Bloch model

Here are the slides for todays lecture.
They need to be read in parallel with chapter 12 of Ashcroft and Mermin.

A good clip to watch for the introduction to quasicrystal

http://www.youtube.com/watch?v=k_VSpBI5EGM

Types of Quasicrystals

quasiperiodic in two dimensions (polygonal or dihedral quasicrystals)

There is one periodic direction perpendicular to the quasiperodic layers.

• octagonal quasicrystals with local 8-fold symmetry [primitive & body-centered lattices]
• decagonal quasicrystals with local 10-fold symmetry [primitive lattice]
• dodecagonal quasicrystals with local 12-fold symmetry [primitive lattice]

quasiperiodic in three dimensions, no periodic direction

• icosahedral quasicrystals (axes:12x5-fold, 20x3-fold, 30x2-fold) [primitive, body-centered & face-centered lattices]

new type (reported in Nature, Nov.2000)

• "icosahedral" quasicrystal with broken symmetry (stable binary Cd_5.7Yb)

Types of Quasicrystal

- In two dimensions, there are 3 types:

* octagonal quasicrystals with local 8-fold symmetry [primitive & body-centered lattices]
* decagonal quasicrystals with local 10-fold symmetry [primitive lattice]
* dodecagonal quasicrystals with local 12-fold symmetry [primitive lattice]
- In three dimensions (no periodic direction), there is only one type:

* icosahedral quasicrystals (axes:12x5-fold, 20x3-fold, 30x2-fold) [primitive, body-centered & face-centered lattices]
- Interestingly, there is a New type that was reported in Nature, Nov.2000). it is "icosahedral" quasicrystal with broken symmetry (stable binary Cd5.7Yb).

The reciprocal lattice and first Brillouin zone

The link to the PDF helps me for better understanding of reciprocal lattice and first Brillouin zone in 2D and 3D.

http://www.chem.tamu.edu/rgroup/hughbanks/courses/673/handouts/translation_groups3.pdf

I hope it can be useful.

A site showing 2D reciprocal vectors

This page shows 2d reciprocal vectors.

http://www.matter.org.uk/diffraction/geometry/2D_reciprocal_lattices.htm

This should be helpful for anyone who hasn't figured this out yet.

It's not a perfect site though :P

Planes Groups and some Folds Numbers

Cited in : http://users.aber.ac.uk/ruw/teach/334/groups.php

General form of reciprocal basis vectors

A general form of a set of reciprocal basis vectors, b_i, is

b_i ·a_j = 2πδ_ij

Where a_j is the real-space basis.

Why 5 fold symmetry is a problem?

As we know from the lecture that the symmetry axis of 1,2,3,4,6 fold exist in a periodic lattice . However, when we try 5 or 7 fold symmetry to construct a periodic lattice,it is hard to do it. the reason for this ,is can not fill the area of plane with a connected array of pentagons.

The definition of quasicrysta

As i found in one website that the Quasicrystal is a form of solid matter whose atoms are arranged like those of a crystal but assume patterns that do not exactly repeat themselves.

Penrose and Aperiodic Tiling

The tiling shown in the lectures was in fact one of three forms of Penrose tiling, and was not in fact the original form. The original created by Penrose used 5 different tile styles, shown here.

http://en.wikipedia.org/wiki/File:Penrose_Tiling_(P1).svg

The symmetry and other properties are completely equivalent between the three styles, and is an example of of substitution. This is where one can break the original shapes into smaller variations, and use these to reconstruct the the tiling.

Due to this possible substitution the tiling shares some features with fractals, in that it may be inflated or deflated, yet present the same image and symmetry.

A list of other forms of aperiodic tiling in various geometries can be found here:
http://en.wikipedia.org/wiki/List_of_aperiodic_sets_of_tiles

Mid-semester exam

Details and some sample questions are here.
The exam is worth 20% of the summative assessment.
We will look at some of the sample questions at next wednesday's tutorial.

Two helpful websites

A tutorial website from the University of Cambridge helps to understand how to generate the reciprocal lattice and to construct Brillouin Zones.

http://www.doitpoms.ac.uk/tlplib/brillouin_zones/aims.php

Self study material in Solid State Electronics using Multimedia: A great website that shows lots of plots for the Crystal structure, Brillouin zones, energy bands and energy surface...
http://people.seas.harvard.edu/~jones/ap216/lectures/ls_2/ls2_u7/sse_tut_1/solid1.html

The advantage of Bloch model over Sommerfeld model

Firstly, the Bloch model can explain the existance of the insulators. Secondly, the sign of the Hall effect can be given by the Bloch model. The Sommerfeld model cannot explain these, however, many of the problems of Sommerfeld model can be solved by taking the interaction of the electrons with the periodic potential.

Next tuesdays lecture

Paul Shaw will give this. Some background reading is here.

Reading and some notes for next week

Ashcroft and Mermin
Chapter 12
Take particular note of
Table 12.1
Equations 12.6
subsection: Filled bands are inert
subsection: Holes

Here are some preliminary notes.

Tight-Binding Model

http://www.cmmp.ucl.ac.uk/~ahh/teaching/3C25/Lecture20s.pdf

For anyone interested this collection of slides is a useful discussion of the basics of tight-binding theory. It goes through the same mathematical derivation of the energy eigenvalues, but also gives a good graphical example in k-space.

Wednesday's Tutorial

This is just a reminder that during tomorrow's tutorial (now at noon) we will be looking at the "Bloch" model of "Solid State Simulations".

Please bring with you :

• A laptop with the software installed
• A copy of the "Bloch" chapter from "Simulations for Solid State Physics". You can download it here if you haven't already done so.

Schedule for wednesday April 6

Just for this week the tutorial and lecture time slots will be swapped.

noon - Paul will give the tutorial

2pm - Ross will give the lecture

Tomorrows lecture on quasi-crystals

Here is the current version of the slides that will go with my lecture on Quasi-crystals.

Nearly free Electron Metal

Here are some slides, easy to understand nearly free electrons metal. http://www.cmmp.ucl.ac.uk/~ahh/teaching/3C25/Lecture19s.pdf

Updated lecture notes

You can download the latest version of my lecture notes here.

six of Brillioun zones step by step

I found in my papers how to find the Brillioun zones in a lattice . so , I want to share you that

Latest Lecture Note

You can download an update version of my lecture notes here.

Bragg plane

A Bragg plane for two points in a lattice is the plane which is perpendicular to the line between the two points and passes through the bisector of that line. The first Brillouin zone for a point in a lattice is the set of points that are closer to the point than the Bragg plane of any point. In other words one can reach any of the points in the first Brillouin zone of a lattice point without crossing the Bragg plane of any other point in the lattice.

The nth Brillouin Zone

I've found an interesting definition for Brillouin zones:
"The nth Brillouin zone is the set of k-space points reached from k=0 by crossing exactly n-1 zone boundaries in the outward direction."

two ways to proof the Bragg's law

There are two ways to proof Bragg's law, one of them is in wikipedia on this link

http://en.wikipedia.org/wiki/Bragg's_law

while the other one which is much easier can be written like that

AB = 2BC = nλ

sin θ = BC/d

BC = d sin θ

2BC = 2d sin θ = nλ

Bragg and Van Laue diffraction

Bragg and Van Laue formulation are two approach for reaching to the same phenomenon. For a constructive interference K = k'-k must be reciprocal lattice vector with K = 2dsina = (2pi/d)n, where a is the angle between the incident ray and lattice plane. So we reach the Bragg condition which is 2dsina=n* wavelength of incident ray.

Ewald's sphere

Ewald's sphere can be used in electron, neutron and X-ray crystallography. The relation between the wavelength of the incident beams and diffracted X-ray beams, the diffraction angle for a given reflection and the reciprocal lattice can be understood by Ewald's sphere.

Fermi surface

In this website you can see Fermi surface for different elements http://www.phys.ufl.edu/fermisurface/

Cleavage (crystal)

Cleavage is an important mechanism in examining the structure of solids and also in some industries. Some crystals have the probability to split along certain crystal planes. And because of the repeated structure these distinct and week planes repeat and become visible even to the naked eye.
Pic. ref.: ( http://www.doitpoms.ac.uk/tlplib/atomic-scale-structure/single2.php?printable=1)

The Energy gap in superconductors

The energy gap in superconductors is the gap between the energy bands which are fully occupied by electrons, and the bands which are fully empty. That Eg is one of the superconductors' properties that don't show in any other materials. The size of the energy band is about 1 eV, which is the required energy to break the band between 2 electrons pairs. At zero temperature, the electrons jump over the energy gap and create holes.

Constructing Brillouin Zones

I read one paper about Brillouin Zones that said they are a significant feature of crystal structures and their constructing for a two dimensional lattice is easier than in a three dimensional lattices. with two dimensional square lattice, Brillouin Zones constructed from the perpendicular bisectors of the vectors joining a reciprocal lattice point to neighbouring lattice point. Below two pictures show the construction of first and second Brillouin Zones

The reduced zone scheme

The reduced zone scheme is the representation of all energy bands within the 1st Brillouin zone. I will here describe in detail how it is constructed. We all know that the graph of E vs k for free electron in 1-D is given by a parabola (E = h_bar^2 k^2 / 2m). One such parabola can be drawn at (centred on) each K. The intersection of two parabolae is the bragg plane (at K/2, ... etc). The energy band is distorted near this Bragg plane (in the presence of the weak periodic potential) such that there is in fact no intersection, the bands are cut off from one-another and reach a value Uk higher and lower than the value which they would have intersected at. This is the origin of the 'band gap' of thickness 2Uk. Each Bragg plane has this effect on the parabola, that is, there is a band gap (of certain thickness) at each bragg plane. The reduced zone scheme arises when we represent all of the bands and their gaps within the first brillouin zone by translating them in with the appropriate reciprocal lattice vector.

Basic concept of Energy Bands

In solid state ,we have three different material metal,semiconductor,and insulator . The band energy of these materials depends on the gap between valence band and conduction band . We clearly can see from the picture that the band gap reduces from insulator to metal (overlap) which means metal does not need heat to be conductor while semiconductor needs.

1st Brillouin Zone

Everything within the lattice may be described as something within the 1st Brillouin zone of the reciprocal lattice, displaced by a vector G. This is how we get the "reflections" in the reduced zone of the energy- k plots seen in the notes and lecture.

Assumption of Weak Potential

There are a number of instances where the assumption of a weak potential gives an accurate description of electrons moving through a lattice, that is, the electrons behave almost as free electrons. There are two reasons why this is sometimes the case.
Firstly, the interaction of electrons with the ions is much stronger at small separations, the conduction electrons are separated from the ions by the core electrons that surround it, and so the interaction is weak. Secondly, in this region in which the conduction electrons are allowed, their mobility can have the effect of screening the fields of the ions, making the potential experienced by any given conduction electron very weak.

Photonic Crystals

Recently we covered Bloch's theorem and its application to the motion of electrons in a periodic potential, i.e. our crystal lattice of ions. The theorem however is not strictly contained to crystal structures, as it only discussed the translationally invariant Hamiltonians and the resulting form of wavefunctions.

An electromagnetic analogue of the above behaviour is found in optical nanostructures which exhibit a periodic potential, due to a periodic dielectric of the material. The behaviour of the EM field in the structure is found to mirror that of the electrons in the crystal structure.

Although we haven't yet covered it, the greatest similarity is between the Bloch treatment of semi-conductors and this photonic crystal. Analogous to the semi-conductor, forbidden energy zones or wavelength band gaps are formed. This gives rise to a range of behaviours such as inhibition of spontaneous emission and and high reflecting omni-directional mirrors.